+theorem nat_compare_to_Prop: \forall n,m:nat.
+match (nat_compare n m) with
+ [ LT \Rightarrow (lt n m)
+ | EQ \Rightarrow (eq nat n m)
+ | GT \Rightarrow (lt m n) ].
+intros.
+apply nat_elim2 (\lambda n,m.match (nat_compare n m) with
+ [ LT \Rightarrow (lt n m)
+ | EQ \Rightarrow (eq nat n m)
+ | GT \Rightarrow (lt m n) ]).
+intro.elim n1.simplify.reflexivity.
+simplify.apply le_S_S.apply le_O_n.
+intro.simplify.apply le_S_S. apply le_O_n.
+intros 2.simplify.elim (nat_compare n1 m1).
+simplify. apply le_S_S.apply H.
+simplify. apply le_S_S.apply H.
+simplify. apply eq_f. apply H.
+qed.
+
+theorem nat_compare_n_m_m_n: \forall n,m:nat.
+eq compare (nat_compare n m) (compare_invert (nat_compare m n)).
+intros.
+apply nat_elim2 (\lambda n,m.eq compare (nat_compare n m) (compare_invert (nat_compare m n))).
+intros.elim n1.simplify.reflexivity.
+simplify.reflexivity.
+intro.elim n1.simplify.reflexivity.
+simplify.reflexivity.
+intros.simplify.elim H.reflexivity.
+qed.
+
+theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
+((lt n m) \to (P LT)) \to ((eq nat n m) \to (P EQ)) \to ((lt m n) \to (P GT)) \to
+(P (nat_compare n m)).
+intros.
+cut match (nat_compare n m) with
+[ LT \Rightarrow (lt n m)
+| EQ \Rightarrow (eq nat n m)
+| GT \Rightarrow (lt m n)] \to
+(P (nat_compare n m)).
+apply Hcut.apply nat_compare_to_Prop.
+elim (nat_compare n m).
+apply (H H3).
+apply (H2 H3).
+apply (H1 H3).
+qed.